Nonlinear Multiscale Phenomena: Analysis, Control, and Computation

用尽可能简单的模型来捕捉非线性现象的本质行为在科学和工程中具有极其重要的意义。这有助于理解基本机制，设计和实现用于模拟和控制设备的有效数值方法，以及分析模型和算法。现代研究的这些关键方面在这个研究项目中混合在一起，该研究项目处理物理和生物现象的建模、公式和数值分析，其规模是表面张力与整体效应竞争，原则上可以被操纵(或控制)以产生科学上有趣和实用的动力学行为。这项工作的应用包括纳米和微技术(如微电子机械系统(MEMS)的设计和控制)、生物技术(如生物膜的研究)和高性能计算(如设计新的高效数值方法)。这项工作的结果将增强建模和预测能力，并帮助学生和博士后在令人兴奋的、具有数学和计算挑战性的、实际相关的研究领域进行教育。该项目研究生物膜、磁流体、液晶和双层致动器等模型，这些模型由定义在预先未知的可变形区域上的非线性几何偏微分方程组控制。数值逼近是通过自适应有限元方法、后验误差估计和多层求解器来实现的，这允许以相对较小的计算资源来解决具有非常不同的时空尺度的问题。该项目将促进对自适应逼近方法和几何在以下关键问题中的作用的理解：1.椭圆型偏微分方程组的自适应有限元方法的收敛和复杂性；分数阶扩散、杂交不连续Galerkin方法、hp-有限元和等几何方法的研究以及参数曲面上的Laplace-Beltrami算子。2.可变形区域和曲面上抛物型偏微分方程组的高阶任意拉格朗日-欧拉方法的设计。3.控制涉及表面张力和磁效应的问题，有或没有自由边界，与技术和生物医学中的设备设计有关。4.磁流体和液晶的计算建模和分析；它们是技术上有用的和数学上有趣的复杂流体，可以由磁场和电场驱动，从而为特定目的进行操纵和控制。5.几何偏微分方程组的新型有限元方法：处理具有等距约束的大变形，这是双层驱动器的典型特征，并处理完全非线性偏微分方程组。

Capturing the essential behavior of nonlinear phenomena with the simplest possible models is of paramount importance in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of devices, and the analysis of both models and algorithms. These crucial aspects of modern research are blended together in this research project, which deals with modeling, formulation, and numerical analysis of physical and biological phenomena at a scale where surface tension competes with bulk effects and could in principle be manipulated (or controlled) to produce scientifically interesting and practically useful dynamical behavior. Applications of the work include nano and microtechnology (such as the design and control of micro electro-mechanical systems (MEMS)), biotechnology (such as the study of biomembranes), and high performance computing (such as the design of novel efficient numerical methods). Results of the work will enhance modeling and prediction capabilities and help educate students and postdocs in exciting, mathematically and computationally challenging, and practically relevant areas of research. This project investigates models, such as biomembranes, ferrofluids, liquid crystals, and bilayer actuators, that are governed by nonlinear geometric partial differential equations defined on deformable domains that are unknown beforehand. Numerical approximation is carried out via adaptive finite element methods, with a posteriori error estimation and multilevel solvers, which allow for the resolution of problems with very disparate space-time scales with relatively modest computational resources. The project will advance understanding of adaptive approximation methods and the role of geometry in key questions concerning:1. Convergence and complexity of adaptive finite element methods (FEM) for elliptic PDE; study of fractional diffusion, hybridizable discontinuous Galerkin methods, hp-FEM and isogeometric methods, and the Laplace-Beltrami operator on parametric surfaces. 2. Design of high order arbitrary Lagrangian-Eulerian methods for parabolic PDE on deformable domains and surfaces. 3. Control of problems involving surface tension and magnetic effects, with or without free boundaries, relevant for device design in technology and biomedicine. 4. Computational modeling and analysis of ferrofluids and liquid crystals; these are technologically useful and mathematically intriguing complex fluids which can be actuated by magnetic and electric fields, and thus manipulated and controlled for specific purposes. 5. Novel FEM for geometric PDE: handling of large deformations with isometry constraints, typical of bilayer actuators, and dealing with fully nonlinear PDE.